Category talk:Paths in Digraphs

Can it be useful to put more explicitly that the sequence of vertices in a path must have more than one vertex?

There is the criteria saying $P$ begins with $v_1$ and ends with $v_n$, and that the sequence is of the form $(v_1, v_2, \ldots, v_n)$, but maybe that doesn't technically rule out the sequence $(v_1)$, if $v_n = v_2$ yields the sequence $(v_1, v_2)$ then why can't $v_n = v_1$ yield the sequence $(v_1)$?

Additionally, for the other constraints:

$P$ begins with $v_1$ and ends with $v_n$. (this criteria is met for the sequence $(v_1)$ if $v_1 = v_n$)

Each arc $e_j$ is incident from $v_j$ and incident to $v_{j+1}$ and all arcs are distinct. (this is met trivially if the sequence of arcs is empty and there is no rule saying that it can't be)

All vertices (except perhaps the first and last ones) are distinct. (this is met for $(v_1)$ as well).

Obviously a single vertex path is absurd given what a path is normally assumed to be, but this is after all a definition.

Please also let me know if i'm missing something or if this kind of thinking is generally a bit over the top, because if it is then I'll just set it aside from now on. I do formal validation of software where this kind of stuff is very important, but in math maybe this is a bit too much since the definition does make perfect sense to me as-is and I would assume that it does for others too. --Androlo (talk) 14:51, 4 March 2024 (UTC)

A single vertex path is still a path, in the same way that $0$ is still a number and $\set {}$ is still a set.

There may be a case to identify a single-vertex path as a "degenerate path", but I have yet to see a source work which goes into this specific detail.

If you can find a source which elaborates on this, you are invited to share it. --prime mover (talk) 10:55, 4 March 2024 (UTC)

Incidentally, please sign your posts. --prime mover (talk) 10:55, 4 March 2024 (UTC)

Message received, and also the edits are now signed. Sorry about that.

I had only heard of empty paths with no vertices and no edges, and they can be concatenated with another path (front or back) without changing that path, like adding $0$ to a number, but i had not heard of paths with one vertex and no edges. But since one vertex and no edge is a valid path then the issue is resolved.

And if I can just ask: when an issue has been resolved, is it customary for the person who opened the page to also delete it, or should it remain? --Androlo (talk) 14:51, 4 March 2024 (UTC)

No rigorous procedure. They get deleted eventually. This can be done only by users with the appropriate level of authorisation. --prime mover (talk) 17:49, 4 March 2024 (UTC)